So, \(\Psi ^\bullet \)  defines a morphism of double complexes between \((\Omega ^\bullet (G^{(\bullet )}),d,\delta )\)  and \((C^\bullet (G^{(\bullet )}),\partial ,\overline{\delta })\)  inducing isomorphisms between the vertical cohomologies of the complexes. Therefore, as consequence of all the facts stated above, by using a usual argument of spectral sequences (see for instance [1]}) we conclude:
